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A New Framework of GPU-Accelerated Spectral Solvers: Collocation and Glerkin Methods for Systems of Coupled Elliptic Equations

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Abstract

Spectral methods are useful for applications that benefit from high-order precisions. However, if the same number of degrees of freedom is used, the computational cost of a spectral method is considerably higher than that of a general finite difference or finite element method. After the investigation in Chen et al. (J Comput Phys 250:555–564, 2013), we provide for the first time a framework of graphics processing units (GPU)-accelerated spectral methods for systems of coupled elliptic equations. The involved dense matrix computations, as the main obstacle for fast spectral methods on a traditional CPU, turns out to be an opportunity for high speedups on a many-core GPU. We obtain an order-of-magnitude speedup for solving 2-D and 3-D systems using a Kepler 20 GPU over a high-end multi-core processor, with two popular spectral methods, namely, the spectral collocation method and the spectral-Galerkin method. The new framework is applicable to systems of \(L\) coupled second-order equations with general boundary conditions, where \(L\) is an integer of moderate size. The ultimate goal is to apply the developed solver to complex and nonlinear time-dependent problems. As two interesting examples, a 2-D FitzHugh-Nagumo equation is solved with the spectral collocation method and a 3-D Cahn-Hilliard equation is solved with the spectral-Galerkin method. We thus demonstrate a practical solution for demanding problems that utilize high-order spatial resolution and longer run-times.

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References

  1. Barkley, D.: A model for fast computer simulation of waves in excitable media. Physica D 49(1), 61–70 (1991)

    Article  Google Scholar 

  2. Bauke, H., Keitel, C.H.: Accelerating the Fourier split operator method via graphics processing units. Comput. Phys. Commun. 182(12), 2454–2463 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258 (1958)

    Article  Google Scholar 

  4. Chen, F., Shen, J.: Efficient spectral-galerkin methods for systems of coupled second-order equations and their applications. J. Comput. Phys. 231(15), 5016–5028 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, F., Shen, J.: A GPU parallelized spectral method for elliptic equations in rectangular domains. J. Comput. Phys. 250, 555–564 (2013)

    Article  MathSciNet  Google Scholar 

  6. Dongarra, J.: Preface: basic linear algebra subprograms technical (blast) forum standard. Int. J. High Perform. Comput. Appl. 16(2), 115–115 (2002)

    Article  Google Scholar 

  7. Dongarra, J., Kurzak, J., Luszczek, P., Tomov, S.: Dense linear algebra on accelerated multicore hardware. In: Michael, W.B., Gallivan, K.A., Gallopoulos, E., Grama, A., Philippe, B., Saad, Y., Saied, F. (eds.) High-Performance Scientific Computing, pp. 123–146. Springer, London (2012)

    Chapter  Google Scholar 

  8. Elling, T.J. : GPU-accelerated Fourier-continuation solvers and physically exact computational boundary conditions for wave scattering problems. PhD thesis, California Institute of Technology (2013)

  9. FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biol. 17(4), 257–278 (1955)

    Google Scholar 

  10. Fitzhugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1(6), 445–466 (1961)

    Article  Google Scholar 

  11. Gottlieb, D., Lustman, L.: The spectrum of the Chebyshev collocation operator for the heat equation. SIAM J. Numer. Anal. 20, 909–921 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gumerov, N.A., Karavaev, A.V., Sharma, A.S., Shao, X., Papadopoulos, K.D.: Efficient spectral and pseudospectral algorithms for 3D simulations of whistler-mode waves in a plasma. J. Comput. Phys. 230(7), 2605–2619 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Keener, J.P., Sneyd, J.: Mathematical Physiology: I: Cellular Physiology. Springer, London (2008)

    Google Scholar 

  14. Khanna, G.: High-precision numerical simulations on a CUDA GPU: kerr black hole tails. J. Sci. Comput. 56(2), 366–380 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Klöckner, A., Warburton, T., Bridge, J., Hesthaven, J.S.: Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys. 228(21), 7863–7882 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lawson, C.L., Hanson, R.J., Kincaid, D.R., Krogh, F.T.: Basic linear algebra subprograms for fortran usage. ACM Trans. Math. Softw. 5(3), 308–323 (1979)

    Article  MATH  Google Scholar 

  17. Leclaire, S., El-Hachem, M., Trpanier, J.-Y. and Reggio, M.: High order spatial generalization of 2D and 3D isotropic discrete gradient operators with fast evaluation on GPUs. J. Sci. Comput., pp. 1–29 (2013)

  18. Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50(10), 2061–2070 (1962)

    Article  Google Scholar 

  19. Olmos, D., Shizgal, B.D.: Pseudospectral method of solution of the FitzhughNagumo equation. Math. Comput. Simul. 79(7), 2258–2278 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Quarteroni, A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)

    Google Scholar 

  21. Ruetsch, G., Micikevicius, P.: Optimizing matrix transpose in CUDA. NVIDIA CUDA SDK Application Note (2009)

  22. Shahbazi, K., Hesthaven, J.S., Zhu, X.: Multi-dimensional hybrid fourier continuationWENO solvers for conservation laws. J. Comput. Phys. 253, 209–225 (2013)

    Article  MathSciNet  Google Scholar 

  23. Shen, J.: Efficient spectral-galerkin method i. direct solvers of second-and fourth-order equations using legendre polynomials. SIAM J. Scient. Comput. 15(6), 1489–1508 (1994)

    Article  MATH  Google Scholar 

  24. Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, New York (2006)

    MATH  Google Scholar 

  25. Shen, J., Yang, X.: Numerical approximations of allen-cahn and cahn-hilliard equations. Discret. Contin. Dyn. Syst. Ser. A 28, 1669–1691 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shen, J.: Spectral Methods: Algorithms, Analysis and Applications (Springer Series in Computational Mathematics). Springer, Berlin (2010)

    Google Scholar 

  27. Wadbro, E., Berggren, M.: Megapixel topology optimization on a graphics processing unit. SIAM Rev. 51(4), 707–721 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Zhang, J., Yan, G.: A lattice boltzmann model for the reaction-diffusion equations with higher-order accuracy. J. Sci. Comput. 52(1), 1–16 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The author first would like to thank the referees for helpful suggestions and comments. The author would also like to thank Jie Shen at Purdue University, Jan Hesthaven at the EPFL, Kevin Wang at Virginia Tech, and Michael Frank at Brown University for their help and insightful suggestions. The author acknowledges partial support by OSD/AFOSR FA9550-09-1-0613 and AFOSR FA9550-12-1-0463. The computing facility is provided by Division of Applied Mathematics and the Center for Computation and Visualization (CCV) at Brown University.

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  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912 , USA

    Feng Chen

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  1. Feng Chen
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Correspondence to Feng Chen.

Appendix: Sample CUDA Codes

Appendix: Sample CUDA Codes

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Chen, F. A New Framework of GPU-Accelerated Spectral Solvers: Collocation and Glerkin Methods for Systems of Coupled Elliptic Equations. J Sci Comput 62, 575–600 (2015). https://doi.org/10.1007/s10915-014-9868-3

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